Multiple Regressioin Analyse (a)

*The author of this computation has been verified*

R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Wed, 18 Nov 2009 10:47:31 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Nov/18/t1258566577b3y662y6rpzaiwn.htm/, Retrieved Wed, 18 Nov 2009 18:49:49 +0100

BibTeX entries for LaTeX users:

@Manual{KEY, author = {{YOUR NAME}}, publisher = {Office for Research Development and Education}, title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2009/Nov/18/t1258566577b3y662y6rpzaiwn.htm/}, year = {2009}, } @Manual{R, title = {R: A Language and Environment for Statistical Computing}, author = {{R Development Core Team}}, organization = {R Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2009}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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4 7.2 102.9 271244 4.1 7.4 97.4 269907 4 8.8 111.4 271296 3.8 9.3 87.4 270157 4.7 9.3 96.8 271322 4.3 8.7 114.1 267179 3.9 8.2 110.3 264101 4 8.3 103.9 265518 4.3 8.5 101.6 269419 4.8 8.6 94.6 268714 4.4 8.5 95.9 272482 4.3 8.2 104.7 268351 4.7 8.1 102.8 268175 4.7 7.9 98.1 270674 4.9 8.6 113.9 272764 5 8.7 80.9 272599 4.2 8.7 95.7 270333 4.3 8.5 113.2 270846 4.8 8.4 105.9 270491 4.8 8.5 108.8 269160 4.8 8.7 102.3 274027 4.2 8.7 99 273784 4.6 8.6 100.7 276663 4.8 8.5 115.5 274525 4.5 8.3 100.7 271344 4.4 8 109.9 271115 4.3 8.2 114.6 270798 3.9 8.1 85.4 273911 3.7 8.1 100.5 273985 4 8 114.8 271917 4.1 7.9 116.5 273338 3.7 7.9 112.9 270601 3.8 8 102 273547 3.8 8 106 275363 3.8 7.9 105.3 281229 3.3 8 118.8 277793 3.3 7.7 106.1 279913 3.3 7.2 109.3 282500 3.2 7.5 117.2 280041 3.4 7.3 92.5 282166 4.2 7 104.2 290304 4.9 7 112.5 283519 5.1 7 122.4 287816 5.5 7.2 113.3 285226 5.6 7.3 100 287595 6.4 7.1 110.7 289741 6.1 6.8 112.8 289148 7.1 6.4 109.8 288301 7.8 6.1 117.3 etc...

Output produced by software:

Enter (or paste) a matrix (table) containing all data (time) series. Every column represents a different variable and must be delimited by a space or Tab. Every row represents a period in time (or category) and must be delimited by hard returns. The easiest way to enter data is to copy and paste a block of spreadsheet cells. Please, do not use commas or spaces to seperate groups of digits! Summary of computational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Cons.index[t] = + 31.0503934631203 -0.855669637876198Werkl.graad[t] + 0.00418233696733515Industr.prod.[t] -7.5847943119858e-05BrutoSchuld[t] + 0.0336106524518894t + e[t] Multiple Linear Regression - Ordinary Least Squares Variable Parameter S.D. T-STAT H0: parameter = 0 2-tail p-value 1-tail p-value (Intercept) 31.0503934631203 6.889019 4.5072 3.1e-05 1.6e-05 Werkl.graad -0.855669637876198 0.304314 -2.8118 0.006647 0.003323 Industr.prod. 0.00418233696733515 0.019547 0.214 0.831299 0.41565 BrutoSchuld -7.5847943119858e-05 2e-05 -3.7484 0.000402 0.000201 t 0.0336106524518894 0.02078 1.6174 0.111032 0.055516 Multiple Linear Regression - Regression Statistics Multiple R 0.643250979798282 R-squared 0.413771823011449 Adjusted R-squared 0.374689944545546 F-TEST (value) 10.5873064257247 F-TEST (DF numerator) 4 F-TEST (DF denominator) 60 p-value 1.47751398660301e-06 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.21493556878008 Sum Squared Residuals 88.5641061772192 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 4 4.78024571319966 -0.780245713199663 2 4.1 4.72112828470714 -0.621128284707141 3 4 3.51000136868156 0.489998631318439 4 3.8 3.10179192219282 0.698208077807177 5 4.7 3.08635368840303 1.61364631159697 6 4.3 4.01995858146111 0.280041418538888 7 3.9 4.69897114129815 -0.79897114129815 8 4 4.51277133797063 -0.512771337970634 9 4.3 4.06974586171185 0.230254138288152 10 4.8 4.04198599150427 0.758014008495728 11 4.4 3.88080559612569 0.519194403874309 12 4.3 4.52124955828112 -0.221249558281123 13 4.7 4.64582997227179 0.05417002772821 14 4.7 4.64137355869592 0.0586264413040822 15 4.9 3.98357418759786 0.916425812402138 16 5 3.80611566695485 1.19388433304515 17 4.2 4.07349634563290 0.126503654367104 18 4.3 4.3125218277679 -0.0125218277679028 19 4.8 4.42809440395341 0.371905596046585 20 4.8 4.48922048211549 0.310779517884513 21 4.8 3.95536007754011 0.84463992245989 22 4.2 3.99360006817792 0.206399931822081 23 4.6 3.90152142901983 0.698478570980173 24 4.8 4.24476053476615 0.555239465233848 25 4.5 4.62887883474099 -0.128878834740988 26 4.4 4.97503705762967 -0.575037057629668 27 4.3 4.88121456422179 -0.58121456422179 28 3.9 4.64215329408299 -0.742153294082993 29 3.7 4.73330448695077 -1.03330448695077 30 4 5.06914306819504 -1.06914306819504 31 4.1 5.0876507301057 -0.987650730105703 32 3.7 5.31380078979424 -1.61380078979424 33 3.8 4.99280896508345 -1.19280896508345 34 3.8 4.90540910069902 -1.10540910069902 35 3.8 4.57673504672031 -0.776735046720307 36 3.3 4.84185381700343 -1.54185381700343 37 3.3 4.91825204191893 -1.61825204191893 38 3.3 5.19686236275331 -1.89686236275332 39 3.2 5.19332267801602 -1.99332267801602 40 3.4 5.13358665582028 -1.73358665582028 41 4.2 4.85558098104344 -0.655580981043442 42 4.9 5.43853332439245 -0.538533324392449 43 5.1 5.18763050123493 -0.0876305012349274 44 5.5 5.20849413238926 0.291505867610741 45 5.6 4.92122896213703 0.678771037862972 46 6.4 5.00795486177943 1.39204513822057 47 6.1 5.35202714349566 0.747972856504344 48 7.1 5.77960184801854 1.32039815198146 49 7.8 5.96065883254409 1.83934116745591 50 7.9 5.65616137387511 2.24383862612489 51 7.4 4.799339975313 2.60066002468700 52 7.5 4.49318341058672 3.00681658941328 53 6.8 4.47658947290765 2.32341052709235 54 5.2 4.2908651353457 0.9091348646543 55 4.7 4.29616626278227 0.403833737217734 56 4.1 3.9150241187067 0.184975881293299 57 3.9 2.35383124895949 1.54616875104051 58 2.6 1.82879296214397 0.77120703785603 59 2.7 2.24234950189386 0.457650498106138 60 1.8 2.4089452095133 -0.608945209513298 61 1 2.99520545841159 -1.99520545841159 62 0.3 2.78193388581127 -2.48193388581127 63 1.3 2.33770781604115 -1.03770781604115 64 1 1.83381352358225 -0.833813523582252 65 1.1 1.96788896332402 -0.867888963324024 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.0172046959879321 0.0344093919758641 0.982795304012068 9 0.00627686151774557 0.0125537230354911 0.993723138482254 10 0.00210292192854465 0.0042058438570893 0.997897078071455 11 0.00110774823653811 0.00221549647307622 0.998892251763462 12 0.000279070206419644 0.000558140412839288 0.99972092979358 13 7.302159868767e-05 0.00014604319737534 0.999926978401312 14 1.47394775945462e-05 2.94789551890923e-05 0.999985260522405 15 3.09325301297538e-06 6.18650602595075e-06 0.999996906746987 16 7.0737454387759e-07 1.41474908775518e-06 0.999999292625456 17 1.58266287937083e-06 3.16532575874166e-06 0.99999841733712 18 9.08508717730247e-07 1.81701743546049e-06 0.999999091491282 19 2.27512792095077e-07 4.55025584190154e-07 0.999999772487208 20 5.6070071653084e-08 1.12140143306168e-07 0.999999943929928 21 2.00824708413005e-08 4.0164941682601e-08 0.99999997991753 22 6.38891300961908e-08 1.27778260192382e-07 0.99999993611087 23 4.39128555472575e-08 8.7825711094515e-08 0.999999956087144 24 2.64051840185015e-08 5.28103680370031e-08 0.999999973594816 25 1.48716415362895e-08 2.97432830725789e-08 0.999999985128359 26 7.11718967398156e-09 1.42343793479631e-08 0.99999999288281 27 3.93871439778569e-09 7.87742879557139e-09 0.999999996061286 28 1.60931078021616e-08 3.21862156043231e-08 0.999999983906892 29 5.69723635889324e-08 1.13944727177865e-07 0.999999943027636 30 3.04788362032232e-08 6.09576724064463e-08 0.999999969521164 31 1.19460702277785e-08 2.38921404555569e-08 0.99999998805393 32 8.62191959637052e-09 1.72438391927410e-08 0.99999999137808 33 4.30867803534006e-09 8.61735607068013e-09 0.999999995691322 34 2.08174394546268e-09 4.16348789092537e-09 0.999999997918256 35 1.23911734025070e-09 2.47823468050140e-09 0.999999998760883 36 1.86550481412691e-09 3.73100962825383e-09 0.999999998134495 37 1.27877983998987e-09 2.55755967997973e-09 0.99999999872122 38 6.75604726453515e-10 1.35120945290703e-09 0.999999999324395 39 2.59541675438056e-09 5.19083350876113e-09 0.999999997404583 40 4.79013685869854e-09 9.58027371739708e-09 0.999999995209863 41 1.46146768175156e-08 2.92293536350313e-08 0.999999985385323 42 7.89830856095394e-07 1.57966171219079e-06 0.999999210169144 43 1.04472555358398e-05 2.08945110716795e-05 0.999989552744464 44 0.000448899010543961 0.000897798021087922 0.999551100989456 45 0.00891045988041258 0.0178209197608252 0.991089540119587 46 0.0884325874063617 0.176865174812723 0.911567412593638 47 0.679551763127902 0.640896473744197 0.320448236872098 48 0.920177569295313 0.159644861409374 0.079822430704687 49 0.937707750893705 0.124584498212591 0.0622922491062954 50 0.952091493983366 0.095817012033268 0.047908506016634 51 0.966349928026503 0.067300143946994 0.033650071973497 52 0.96150840126609 0.0769831974678209 0.0384915987339104 53 0.9471806153874 0.105638769225201 0.0528193846126005 54 0.938549129131858 0.122901741736284 0.0614508708681419 55 0.929466286383505 0.141067427232990 0.0705337136164949 56 0.891355173347351 0.217289653305298 0.108644826652649 57 0.96049803347991 0.0790039330401816 0.0395019665200908 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 35 0.7 NOK 5% type I error level 38 0.76 NOK 10% type I error level 42 0.84 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

R code (references can be found in the software module):

library(lattice) library(lmtest) n25 <- 25 #minimum number of obs. for Goldfeld-Quandt test par1 <- as.numeric(par1) x <- t(y) k <- length(x[1,]) n <- length(x[,1]) x1 <- cbind(x[,par1], x[,1:k!=par1]) mycolnames <- c(colnames(x)[par1], colnames(x)[1:k!=par1]) colnames(x1) <- mycolnames #colnames(x)[par1] x <- x1 if (par3 == 'First Differences'){ x2 <- array(0, dim=c(n-1,k), dimnames=list(1:(n-1), paste('(1-B)',colnames(x),sep=''))) for (i in 1:n-1) { for (j in 1:k) { x2[i,j] <- x[i+1,j] - x[i,j] } } x <- x2 } if (par2 == 'Include Monthly Dummies'){ x2 <- array(0, dim=c(n,11), dimnames=list(1:n, paste('M', seq(1:11), sep =''))) for (i in 1:11){ x2[seq(i,n,12),i] <- 1 } x <- cbind(x, x2) } if (par2 == 'Include Quarterly Dummies'){ x2 <- array(0, dim=c(n,3), dimnames=list(1:n, paste('Q', seq(1:3), sep =''))) for (i in 1:3){ x2[seq(i,n,4),i] <- 1 } x <- cbind(x, x2) } k <- length(x[1,]) if (par3 == 'Linear Trend'){ x <- cbind(x, c(1:n)) colnames(x)[k+1] <- 't' } x k <- length(x[1,]) df <- as.data.frame(x) (mylm <- lm(df)) (mysum <- summary(mylm)) if (n > n25) { kp3 <- k + 3 nmkm3 <- n - k - 3 gqarr <- array(NA, dim=c(nmkm3-kp3+1,3)) numgqtests <- 0 numsignificant1 <- 0 numsignificant5 <- 0 numsignificant10 <- 0 for (mypoint in kp3:nmkm3) { j <- 0 numgqtests <- numgqtests + 1 for (myalt in c('greater', 'two.sided', 'less')) { j <- j + 1 gqarr[mypoint-kp3+1,j] <- gqtest(mylm, point=mypoint, alternative=myalt)$p.value } if (gqarr[mypoint-kp3+1,2] < 0.01) numsignificant1 <- numsignificant1 + 1 if (gqarr[mypoint-kp3+1,2] < 0.05) numsignificant5 <- numsignificant5 + 1 if (gqarr[mypoint-kp3+1,2] < 0.10) numsignificant10 <- numsignificant10 + 1 } gqarr } bitmap(file='test0.png') plot(x[,1], type='l', main='Actuals and Interpolation', ylab='value of Actuals and Interpolation (dots)', xlab='time or index') points(x[,1]-mysum$resid) grid() dev.off() bitmap(file='test1.png') plot(mysum$resid, type='b', pch=19, main='Residuals', ylab='value of Residuals', xlab='time or index') grid() dev.off() bitmap(file='test2.png') hist(mysum$resid, main='Residual Histogram', xlab='values of Residuals') grid() dev.off() bitmap(file='test3.png') densityplot(~mysum$resid,col='black',main='Residual Density Plot', xlab='values of Residuals') dev.off() bitmap(file='test4.png') qqnorm(mysum$resid, main='Residual Normal Q-Q Plot') qqline(mysum$resid) grid() dev.off() (myerror <- as.ts(mysum$resid)) bitmap(file='test5.png') dum <- cbind(lag(myerror,k=1),myerror) dum dum1 <- dum[2:length(myerror),] dum1 z <- as.data.frame(dum1) z plot(z,main=paste('Residual Lag plot, lowess, and regression line'), ylab='values of Residuals', xlab='lagged values of Residuals') lines(lowess(z)) abline(lm(z)) grid() dev.off() bitmap(file='test6.png') acf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Autocorrelation Function') grid() dev.off() bitmap(file='test7.png') pacf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Partial Autocorrelation Function') grid() dev.off() bitmap(file='test8.png') opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0)) plot(mylm, las = 1, sub='Residual Diagnostics') par(opar) dev.off() if (n > n25) { bitmap(file='test9.png') plot(kp3:nmkm3,gqarr[,2], main='Goldfeld-Quandt test',ylab='2-sided p-value',xlab='breakpoint') grid() dev.off() } load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Estimated Regression Equation', 1, TRUE) a<-table.row.end(a) myeq <- colnames(x)[1] myeq <- paste(myeq, '[t] = ', sep='') for (i in 1:k){ if (mysum$coefficients[i,1] > 0) myeq <- paste(myeq, '+', '') myeq <- paste(myeq, mysum$coefficients[i,1], sep=' ') if (rownames(mysum$coefficients)[i] != '(Intercept)') { myeq <- paste(myeq, rownames(mysum$coefficients)[i], sep='') if (rownames(mysum$coefficients)[i] != 't') myeq <- paste(myeq, '[t]', sep='') } } myeq <- paste(myeq, ' + e[t]') a<-table.row.start(a) a<-table.element(a, myeq) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable1.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/ols1.htm','Multiple Linear Regression - Ordinary Least Squares',''), 6, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Variable',header=TRUE) a<-table.element(a,'Parameter',header=TRUE) a<-table.element(a,'S.D.',header=TRUE) a<-table.element(a,'T-STAT<br />H0: parameter = 0',header=TRUE) a<-table.element(a,'2-tail p-value',header=TRUE) a<-table.element(a,'1-tail p-value',header=TRUE) a<-table.row.end(a) for (i in 1:k){ a<-table.row.start(a) a<-table.element(a,rownames(mysum$coefficients)[i],header=TRUE) a<-table.element(a,mysum$coefficients[i,1]) a<-table.element(a, round(mysum$coefficients[i,2],6)) a<-table.element(a, round(mysum$coefficients[i,3],4)) a<-table.element(a, round(mysum$coefficients[i,4],6)) a<-table.element(a, round(mysum$coefficients[i,4]/2,6)) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable2.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Regression Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple R',1,TRUE) a<-table.element(a, sqrt(mysum$r.squared)) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'R-squared',1,TRUE) a<-table.element(a, mysum$r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Adjusted R-squared',1,TRUE) a<-table.element(a, mysum$adj.r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (value)',1,TRUE) a<-table.element(a, mysum$fstatistic[1]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF numerator)',1,TRUE) a<-table.element(a, mysum$fstatistic[2]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF denominator)',1,TRUE) a<-table.element(a, mysum$fstatistic[3]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'p-value',1,TRUE) a<-table.element(a, 1-pf(mysum$fstatistic[1],mysum$fstatistic[2],mysum$fstatistic[3])) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Residual Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Residual Standard Deviation',1,TRUE) a<-table.element(a, mysum$sigma) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Sum Squared Residuals',1,TRUE) a<-table.element(a, sum(myerror*myerror)) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable3.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Actuals, Interpolation, and Residuals', 4, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Time or Index', 1, TRUE) a<-table.element(a, 'Actuals', 1, TRUE) a<-table.element(a, 'Interpolation<br />Forecast', 1, TRUE) a<-table.element(a, 'Residuals<br />Prediction Error', 1, TRUE) a<-table.row.end(a) for (i in 1:n) { a<-table.row.start(a) a<-table.element(a,i, 1, TRUE) a<-table.element(a,x[i]) a<-table.element(a,x[i]-mysum$resid[i]) a<-table.element(a,mysum$resid[i]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable4.tab') if (n > n25) { a<-table.start() a<-table.row.start(a) a<-table.element(a,'Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'p-values',header=TRUE) a<-table.element(a,'Alternative Hypothesis',3,header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'breakpoint index',header=TRUE) a<-table.element(a,'greater',header=TRUE) a<-table.element(a,'2-sided',header=TRUE) a<-table.element(a,'less',header=TRUE) a<-table.row.end(a) for (mypoint in kp3:nmkm3) { a<-table.row.start(a) a<-table.element(a,mypoint,header=TRUE) a<-table.element(a,gqarr[mypoint-kp3+1,1]) a<-table.element(a,gqarr[mypoint-kp3+1,2]) a<-table.element(a,gqarr[mypoint-kp3+1,3]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable5.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Description',header=TRUE) a<-table.element(a,'# significant tests',header=TRUE) a<-table.element(a,'% significant tests',header=TRUE) a<-table.element(a,'OK/NOK',header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'1% type I error level',header=TRUE) a<-table.element(a,numsignificant1) a<-table.element(a,numsignificant1/numgqtests) if (numsignificant1/numgqtests < 0.01) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'5% type I error level',header=TRUE) a<-table.element(a,numsignificant5) a<-table.element(a,numsignificant5/numgqtests) if (numsignificant5/numgqtests < 0.05) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'10% type I error level',header=TRUE) a<-table.element(a,numsignificant10) a<-table.element(a,numsignificant10/numgqtests) if (numsignificant10/numgqtests < 0.1) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable6.tab') }

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